Week 9- Options Flashcards
What does the value of a derivative come from?
The value of a derivative is derived from the value of the underlying asset.
What are options?
Options are special type of financial asset which give the holder the right but not the obligation to buy/sell a particular security at a predetermined price.
What do options allow?
The hedging against risk
When writing an option, what does this do for the risk?
Transfers it from you to a different part
What is the right to buy called?
A call option
What is the right to sell called?
A put option
What is the price at which an option can be exercised called?
The exercise/strike price
The option of whether to buy or sell has a value. What does the buyer of an option pay for this privilege?
A premium
If it is profitable to exercise the option at the asset’s current price, then it is the..
The money option
If it is unprofitable to exercise the option at the asset’s current price, then it is the..
the out of money option
If an option ca only be exercised on a particular day, what is this called?
A European Call
If an option can be exercised on or before a specific date, what is this called?
An American Call
Are premiums higher for American or European calls?
American, as you get more flexibility.
How do we calculate the value of a call option?
Value of the call option CT with stock price of ST and exercise price X is:
CT = max[0,ST – X]
- max refers to the maximum between (ST -X) and 0. (ie the contract value cannot be less than 0, as 0 is worthless)
In which case is the call worth the difference between the current stock price and the exercise price.
- The call is worth the difference between the current stock price and the exercise price. π depends upon whether CT >a : a is the
premium. - If ST <X then -π given +a (then call option has no value and is not exercised)
- If ST >X and a< CT then +π (then call in the money at termination)
In which case does exercising a sell call turn a profit?
If S-X-a>0 (ie C-a>0)
where S = share price
X= exercise price
a= premium
Give one key difference between a short and an option.
In a short you have an obligation to repay, in options there are no obligations.
What is the payoff from buying a call option?
a-C < 0 is where profit is made. This is the mirror image from buying a call option.
With a call option, when will a rational investor exercise? When they make profit?
No, but when the payoff is greater than a premium (ie potentially between the negative payoff of the premium and 0)
How do we calculate the value of a put option?
Value of the put option CT with stock price of ST and exercise price X is:
CT = max[0,X- ST]
What is a put worth, and when is it in profit?
- The put is worth the difference between the current stock price and the exercise price. π depends upon whether PT >b where b is the
premium. - If X> ST and b<P then +π .
- If X<ST then -π given +b.
When would the value of a put contract be zero?
If the stock price is higher than the specified exercise/strike price, as naturally you would rather sell for that instead.
Why are we dealing with European auctions instead of American?
As they are easier to value as there is only a specific date of termination, unlike an American one where there is flexibility.
What are interest rate options used for?
To give the holder a right (but not obligation), to lend or borrow a stated amount at a predetermined rate.
What are currency options used for?
To give the holder the right, but not the obligation to buy or sell a certain amount of currency at a given exchange rate.
What are options also commonly traded by?
Speculators purely for profit.
Does adding T-Bills to a portfolio on the Capital Market Line alter the shape of the distribution?
No, it merely squeezes te distribution as it lowers the variance.
Can options modify the shape of the return distribution?
Yes
What does combining a portfolio with a put do?
Eliminates risk below r*, it effectively eliminates negative outcomes
An option premium consists of which two components?
Intrinsic value and time value
What is the intrinsic value for a call option?
Cash price - Strike price
What is the intrinsic value for a put option?
Strike price - cash price
Describe in/out/at the-money for call options.
- If an intrinsic value for an option exists then in-the-money.
- If the strike price is above the cash price for a call option then it has zero intrinsic value and is out-of-the-money.
- If the strike price=cash price then at-the-money.
What does the intrinsic value reflect the price of?
The intrinsic value reflects the price that could be received if the option were “locked in” today at the current market price.
What does time value equal, if option premium = time value + intrinsic value? Also describe what it reflects.
- Time value = option premium – intrinsic value
- Time value reflects the fact that an option may have more ultimate value than the intrinsic value alone.
- Hence, even if option is out-of-the-money a buyer can hope that at some time prior to expiration changes in the spot price will
move the option into the money.
Why could time value lead to an out-of-money option being in money?
As the price of the stock may change (ie volatility)
What is the upper bound on the European call option?
Upper bound: The value of a call option Ct is below the value of the stock St for all t ≤ T
Ct < St
What is the lower bound on the European call option?
Lower bound: The value of a call option Ct is at least the value of the stock St less the discounted value of the cost of exercising the
call (r is risk-free rate of interest) for all t < T
Ct > St – Xe^-r(T – t)
What iss Xe^-r(T – t)?
The present value of X upon continuoiuus discounting
To understand the upper and lower bounds, we need what?
The no-arbitrage principle- ie an investor cannot make money out of nothing. A portfolio that makes a positive return in all states of the world (return with no downside risk below zero) must cost something to construct.
What is the no-arbitrage principle also sometimes known as?
The “no magic money pump” argument.
Explain why the upper bound is Ct<St
- Time t < T: Suppose Ct ≥ St. Investor A sells a call option on the stock to B and uses some of the proceeds to buy the stock. So, investor A makes a sure gain Ct – St ≥ 0 (constructing this portfolio costs the investor less than nothing).
- Time T: If investor B exercises the call option, A sells the stock to investor B, making X. If investor B does not exercise the call option, A sells the stock to investor C, making ST. Either way, it’s a sure gain of at least min[X, ST] > 0 to investor A.
- Conclusion: Ct ≥ St leads to a money pump. So, Ct < St
Explain why the lower bound for all t < T is
Ct > St – Xe^-r(T – t)
- Time t < T: Investor simultaneously (i) short-sells the stock; (ii)buys a call option in the stock; and (iii) buys Xe^-r(T – t) of the riskfree asset. The value of the resulting portfolio is
Vt = Ct – St + Xe-r(T – t) - Time T: If investor exercises the call option (ST > X) it is worth ST– X. The risk-free assets have earned interest and are now worth X. The short-sold stock will now cost ST to buy back. Value of the portfolio if the call is exercised is therefore now
VT = ST – X – ST + X = 0, but if the option has no value is Vt= -St+X≥ 0 - Either way, the portfolio always has a (weakly) positive value.
If Vt ≤ 0 then the portfolio costs less than nothing to construct, but yields a guaranteed (weakly) positive return. To avoid the money pump problem, the original portfolio must
have cost money to construct (Vt > 0). So
Ct > St – Xe-r(T – t)
What are the 6 underlying assumptions for the Black and Scholes formula to evaluate European options?
- Underlying asset pays no dividends or interest during lifetime;
- Option is European, i.e. cannot be exercised prior to maturity;
- Risk free rate is fixed over the lifetime of the asset;
- Financial markets are perfect, i.e. no transaction costs or taxes;
- Price of underlying asset is log normally distributed; (ie there are no negative values)
- Price moves in continuous time, assuming that stock prices will only move slightly ↑↓ during the next microsecond.
Is the value of St – Xe^-r(T – t) at time T certain?
No, look at the expected value upon expiration.
What is the Black-Scholes model formula?
Ct = StN(d1) – Xe^-r(T – t) N(d2)
where:
-𝐶(𝑡) is the call value of the option with t left before expiration;
- 𝑡 is time until the option expires;
- 𝑆 is current/spot stock price
- 𝑋 is exercise (strike) price of the option
- 𝑒 is 2.71828 – base of the natural system of logs
- 𝑟 is the continuously compounded risk free rate of interest
- 𝑁(𝑑𝑖) is the value of the cumulative normal density function
What does the Black-Scholes state that the cakk value is effectively equal to? What does this mean for each term?
Call Value = (share price×probability1) –
(present value of exercise price×probability2)
* Thus the first term is the expected stock (option) price at period t
* The second term is the expected present value of the exercise price at t.
* The value of the call option is the difference between the two terms
Do increases these variables increase or decrease the call option value?
Price volatility
Time to expiration
Exercise price
Current stock price
Risk free interest rate
Price volatility ↑
Time to expiration ↑
Exercise price ↓
Current stock price ↑
Risk free interest rate ↑
So as the valu of the underlying stock rises, what happens to the intrinsic value of the call option?
It rises
Why does it usually make more sense to sell the option before it expires?
As it should also have time value,as well as just its intrinsic value.
What does an increase in risk do to the value of the option?
It increases it
When does the 45-degree line meet the horizontal axis and what does it represent?
- The 45 degree line meets the horizontal axis at the PV of the exercise price X discounted at the risk-free rate.
- The 45 degree line represents the difference between the share price S and the present value of the exercise price X.
Give 3 practical issues with Black-Scholes
- Quantities such as the stock variance and the risk-free rate assumed to be constant during the life of the option, but in practice these vary
- Large price changes are more frequently observed in the real world than those expected and implied by the Black-Scholes
model - Lognormal prices are a questionable assumption for most stocks
